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Sign-changing blow-up for scalar curvature type equations. (English) Zbl 1277.58010

Summary: Given \((M,g)\) a compact Riemannian manifold of dimension \(n\geq 3\), we are interested in the existence of blowing-up sign-changing families \((u_\varepsilon)_{\varepsilon>0}\in C^{2,\theta}(M)\), \(0\in(0,1)\), of solutions to \[ \Delta_g u_\varepsilon+hu_\varepsilon=| u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon\text{ in }M, \] where \(\Delta_g:=\mathrm{div}_g(\nabla)\) and \(h\in C^{0,\theta}(M)\) is a potential. Assuming the existence of a nondegenerate solution to the limiting equation (which is a generic assumption), we prove that such families exist in two main cases: in small dimension \(n\in\{3,4,5,6\}\) for any potential \(h\) or in dimension \(3\leq n\leq 9\) when \(h\equiv\frac{n-2}{4(n-1)}\mathrm{Scal}_g\). These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of O. Druet [J. Differ. Geom. 63, No. 3, 399–473 (2003; Zbl 1070.53017)] and M. A. Khuri et al. [J. Differ. Geom. 81, No. 1, 143–196 (2009; Zbl 1162.53029)].

MSC:

58J05 Elliptic equations on manifolds, general theory
35J35 Variational methods for higher-order elliptic equations
35J60 Nonlinear elliptic equations
35B44 Blow-up in context of PDEs
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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